How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
4
-
$\begingroup$ You need to specify some extra restrictions on your functions (such as continuity, or linearity) - but until you edit your question it is not clear to readers what tacit conditions you are imposing. $\endgroup$– Yemon ChoiCommented Sep 1 at 23:53
-
1$\begingroup$ Since $\exp:S_d(\mathbb{R})\to S^+_d(\mathbb{R}$) is a diffeomorphism, where $S_d(\mathbb{R})$ is the vector space of symmetric $d$-by-$d$ matrices with real entries (whose dimension is $\tfrac12d(d{+}1)$), and $S^+_d(\mathbb{R})\subset S_d(\mathbb{R})$ is the open cone of positive definite symmetric $d$-by-$d$ matrices, any injection $f:\mathbb{R}^d\to S^+_d(\mathbb{R})$ is uniquely of the form $f = \exp\circ\phi$, where $\phi:\mathbb{R}^d\to S_d(\mathbb{R})$ is an injective map, so you are reduced to 'characterizing' the injective maps $\phi$ into the vector space $S_d(\mathbb{R})$. $\endgroup$– Robert BryantCommented Sep 3 at 12:35
-
$\begingroup$ @RobertBryant What can be said about the topology of space of all semi positive matrix of size $d$ whose matrix norm is $\leq$ 1? It is compact and convex so is it homeomorphic to a disk? $\endgroup$– Ali TaghaviCommented Sep 12 at 15:05
-
$\begingroup$ More generally is every compact convex set in $\mathbb{R}^n$ homeomorphic to a $k$ dimensional disk for some $k\leq n$? $\endgroup$– Ali TaghaviCommented Sep 12 at 15:08
Add a comment
|